https://file.aiquickdraw.com/mathbot/file/1725935428845zlu4fxic.jpg

Let's break down and solve each part of the question from the image.

### Given Information:
- Diane took a loan of **$320,000** five years ago.
- The mortgage term is **30 years**.
- The interest rate was **8% per year compounded monthly**.
- After 5 years, Diane is considering refinancing because the new rate is **6.5% per year compounded monthly**.

We'll tackle the questions one by one.

### (a) What is Diane's current monthly mortgage payment?
To calculate the monthly mortgage payment, we can use the following formula for a fixed-rate mortgage:

\[
M = P \times \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]

Where:
- $ M $ is the monthly mortgage payment
- $ P $ is the loan principal ($320,000)
- $ r $ is the monthly interest rate (annual rate divided by 12, so 8%/12)
- $ n $ is the total number of payments (30 years $\times$ 12 months = 360 months)

### (b) What is Diane's current outstanding principal?
After 5 years of payments, the principal would have been reduced. To find the outstanding principal, we use the amortization formula to determine the remaining balance after 5 years of monthly payments.

### (c) If Diane refinances her property at 6.5% per year for 30 years, what will her new monthly mortgage payment be?
Once we know the outstanding principal from (b), we can treat this as a new loan amount and calculate the monthly payment using the new interest rate of **6.5% compounded monthly** for 30 years.

### (d) How much less will Diane’s monthly mortgage payment be if she refinances?
This will simply be the difference between the old monthly payment and the new one calculated in part (c).

Let me now calculate these parts.

Five years ago, Diane secured a bank loan of $320,000 to help finance the purchase of a loft in the San Francisco Bay area. The term of the mortgage was 30 years, and the interest rate was 8% per year compounded monthly on the unpaid balance. Because the interest rate for a conventional 30-year home mortgage has now dropped to 6.5% per year compounded monthly, Diane is thinking of refinancing her property. (Round your answers to the nearest cent.)

(a) What is Diane's current monthly mortgage payment?
(b) What is Diane's current outstanding principal?
(c) If Diane decides to refinance her property by securing a 30-year home mortgage loan in the amount of the current outstanding principal at the prevailing interest rate of 6.5% per year compounded monthly, what will be her monthly mortgage payment?
(d) How much less will Diane's monthly mortgage payment be if she refinances?

Calculate Diane's monthly mortgage payment, outstanding principal after 5 years, and the potential savings by refinancing her 30-year home loan at a lower interest rate. Includes key formulas for mortgage payments and loan refinancing. Suitable for finance students.

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